BTC

Table Of Statistics (Daily, Returns (%))

BTC (N = 2,081)
Mean (sd) 2,080; 0.22 ± 3.94
median (Q1, Q3) 2,080; 0.19 (-1.21, 1.78)
min -37.16954
max 25.24717
Cumulative Return 466.6306
Normally Distributed VaR -6.25452002497649
Non-Normally Distributed VaR -6.05178991592173

The above statistics table is based on the percentage returns for the daily price valuation of bitcoin; it presents the mean, median, min, and max returns to demonstrate the range as well as the average return over the six year period from 2014 to 2020.

In regards to the mean return (%) of bitcoin, it can be noted that it is positive and relatively close in value to the median price meaning that over a daily basis, bitcoin is expected to provide approximately .22% return of an investment per day during a 6 year period. This can fluctuate depending on the valuation trend as the price of bitcoin can over a larger period be trending downwards, upwards or sideways. Given that the mean and the median are relatively close in value, there is limited skewness in the data, and the skewness is favoring higher returns since the mean is to the right of the median.

Colloquially, a bull market is defined as a financial securitity with a sustained increase in valuation; thus, a bear market is defined as a financial securitity with a sustained decrease in valuation; and a trendless market does not have either a sustained increase or a sustained decrease in valuation.

The standard deviation is approximately 3.94% which means that 95% of the daily returns will most likely fluctuate between [-7.66%, 8.01%]. Anything that occurs outside of this interval would be a rare event with a probability \(\le 5%\), which means that most of the returns would occur be centered around the mean, and 5% of the returns would be \([-\infty,-7.66\%]\cup[8.01\%,\infty]\) (outside of the interval [-7.66%, 8.01%]).

In addition, the minimum provided in this table is the greatest percent decrease in the valuation of bitcoin for a single day, and respectively, the maximum is the greatest percent increase in the valuation of bitcoin for a single day. While the cummulative return is the total percentage return over the full period of the investment.

Finally, the value at risk is approximately 6.25% which means that on a daily time frame, there is a probability of \(5\%\) that one could lose 6.25% on the daily, this is important information for trading and risks could be taken to mitigate the risks of a 6.25% drawback in one’s portfolio through hedging or other means.

Plotting Returns (Daily, Returns)

There two figure presented above is a representation of the returns plotted against time which denotes the returns for each of the change in price valuation on the daily basis. In addition, the histogram represents the frequency for the returns within a .5% interval, and based on the graph there are some outlier in the distribution but it is mostly centered around the mean and it has a gaussian shape meaning that the returns are normally distributed. This can be tested using a form of normality test but will not given the visual shape of the data.

Financial projections of Bitcoin:

BTC With Historical Statistics Used for the Deterministic Projection of Logarithmic Growth

Simple Price Projection with Geometric Brownian Motion

Multiple Price Projections with Geometric Brownian Motion

This data was simulated using the mathematical equation: \[\delta S_t=S_{(t-1)}\cdot(\mu\delta t + \sigma \delta W_t)\] Where in this case \(\delta t = 1\) because the daily variance and the daily mean are being used rather than their annualized counter-parts.

Subsetting the data Into 8 groups:

Major Bull Market

BullMarket (N = 1,068)
Mean (sd) 0.49 ± 3.72
median (Q1, Q3) 0.27 (-0.78, 1.83)
min -21.14486
max 25.24717
Cumulative Return 520.3815
Normally Distributed VaR -5.6378883280451
Non-Normally Distributed VaR -5.35477918110908

Major Bear Market

BearMarket (N = 894)
Mean (sd) 0.01 ± 4.18
median (Q1, Q3) 0.10 (-1.66, 1.63)
min -37.16954
max 18.18776
Cumulative Return 8.210591
Normally Distributed VaR -6.87117808582232
Non-Normally Distributed VaR -6.45995118699981

Minor Bear Market 1

FirstMinorBearMarket (N = 363)
Mean (sd) -0.40 ± 4.39
median (Q1, Q3) -0.01 (-2.38, 1.46)
min -16.8548
max 14.78049
Cumulative Return -143.6408
Normally Distributed VaR -7.61358726528272
Non-Normally Distributed VaR -8.58836887321155

Minor Bull Market 1

FirstMinorBullMarket (N = 194)
Mean (sd) 0.78 ± 3.46
median (Q1, Q3) 0.32 (-0.63, 1.91)
min -8.831367
max 17.35601
Cumulative Return 150.7003
Normally Distributed VaR -4.91756452906168
Non-Normally Distributed VaR -4.72327966403773

Minor Bear Market 2

SecondMinorBearMarket (N = 175)
Mean (sd) -0.25 ± 3.85
median (Q1, Q3) -0.19 (-2.01, 1.35)
min -14.08568
max 15.57634
Cumulative Return -44.38735
Normally Distributed VaR -6.58712229805485
Non-Normally Distributed VaR -6.31733229205686

Minor Bull Market 2

SecondMinorBearMarket (N = 60)
Mean (sd) 0.70 ± 2.71
median (Q1, Q3) 0.12 (-0.96, 1.96)
min -4.211557
max 9.581901
Cumulative Return 41.95282
Normally Distributed VaR -3.75585094901244
Non-Normally Distributed VaR -2.981929208615

Minor Bear Market 3

SecondMinorBearMarket (N = 60)
Mean (sd) 0.70 ± 2.71
median (Q1, Q3) 0.12 (-0.96, 1.96)
min -4.211557
max 9.581901
Cumulative Return 41.95282
Normally Distributed VaR -3.75585094901244
Non-Normally Distributed VaR -2.981929208615

Minor Bull Market 3

SeconMinorBullMarket (N = 2,053)
Mean (sd) 2,052; 0.26 ± 3.86
median (Q1, Q3) 2,052; 0.20 (-1.21, 1.79)
min -21.14486
max 25.24717
Cumulative Return 528.0639
Normally Distributed VaR -6.09523722010413
Non-Normally Distributed VaR -6.04396999141942

For the tables presented above, the shorter term trend changes which occur within the Major Bear market are labeled as “minor.” These minor changes are time frames where a local high was either surpased denoting a bullish change in price valuation or a local low was broken denoting a bearish change in price valuation. The returns of Bitcoin can be compared for each trending period using these tables; in addition, these tables exemplifies the importance of timing an investment correctly and investing in the bottom is not necessary so long as an investment is initialized early in the bullish trend.

Results of 1000 simulated Prices Projections

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
##      30.8    2991.6    9649.9   41609.1   32160.4 1593239.7
Price
5% 494.7718
10% 971.8468
15% 1480.1073
20% 2111.7316
25% 2991.5725
30% 3773.8742
35% 4888.8804
40% 6088.6554
45% 7744.6184
50% 9649.8751
55% 12371.7051
60% 15580.4536
65% 20077.5926
70% 26069.7368
75% 32160.3775
80% 43399.7529
85% 61068.0143
90% 92737.5045

Based on the simulation of 1000 prices using random walk theory, a summary of the statistics can be created in which a cummulative probability of the randomized projections of the price is obtained. This simply means that 95% of the data is \(\le\) to the number given in the simulation.
The distribution of the simulated data can be seen in the histogram given as a exponential distribution. The reason for this is because there is a lower limit occuring in the data.